Electromagnetic wave equation and B5 (group): Difference between pages

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{{originalresearch|article|date=June 2007}}

The '''electromagnetic wave equation''' is a second-order partial differential equation that governs the propagation of electromagnetic waves through free space. The equation, written in terms of either the electric field '''E''' or the magnetic field '''H''', takes the form:

{{Infobox musical artist 2

:<math> {\partial^2 \mathbf{E} \over \partial t^2} \ - \ c^2 \cdot \nabla^2 \mathbf{E} \ \ = \ \ 0</math>

|Name = B5

|Img =

|Img_capt =

|Background = group_or_band

|Origin = {{flagicon|USA}}[[Atlanta, Georgia]], [[United States]]

|Genre = [[Rhythm and blues|R&B]], [[Pop-rap]]

|Years_active = [[2005]]-present

|Label = [[Bad Boy Records]]

|URL = [http://www.b5online.com/ B5Online.com]

|Current_members = [[Dustin Breeding]]<br>[[Kelly Breeding]]<br>[[Patrick Breeding]]<br>[[Carnell Breeding]]<br>[[Bryan Breeding]]

|Former_members =

}}

'''B5''' is an [[African/American]] [[Hip Hop]] [[boy band]] originating from [[St.Petersburg, Florida]]. B5 takes its name from ''Breeding 5'', the last name of the five brothers. B5 consists of Dustin Breeding, Kelly Breeding, Patrick Breeding, Carnell Breeding, and Bryan Breeding, all of which are of African American and White descent. B5 was originally known as the TNT Boyz, featuring only four of the brothers. They renamed themselves B5 after adding the youngest brother, Bryan.

:<math> {\partial^2 \mathbf{H} \over \partial t^2} \ - \ c^2 \cdot \nabla^2 \mathbf{H} \ \ = \ \ 0</math>

B5 is managed by Jim McMahan of JAMM Entertainment Group. Through McMahan's help, B5 landed an audition that eventually got them signed to [[Sean Combs]]' [[Bad Boy Records]].

Since signing a record deal, B5 has toured with [[Mario Winans]], [[Kanye West]], [[Ginuwine]], [[Usher (entertainer)|Usher]], [[Fantasia]], [[Mario]], [[Omarion]], [[T-Pain]], [[Yung Joc]], [[Sean Combs|Diddy]], [[Little Richard]], [[Gladys Knight]], [[Jordan Knight]], [[Snoop Dogg]], [[Boyz II Men]], [[Aly & AJ]], [[Miley Cyrus]], [[Ray J]], [[Pretty Ricky]], [[112 (band)|112]], [[Bobby Valentino]], [[Chris Brown (singer)|Chris Brown]], [[Jonas Brothers]] and many, many others. B5 also performed in the 2005 "Scream Tour IV". On April 27, 2006, B5 joined [[Bow Wow]] on the tour, which featured [[T-Pain]], [[Ray-J]] and others. The tour ended on May 28, 2006. In 2005, they headlined Radio Disney's Jingle Jam Tour. They also had DVDs about their discovery and what they want in the future.

B5 performed a new version of the ''[[High School Musical]]'' song "[[Get'cha Head in the Game]]", produced by [[Andrew Lane]] and [[Eddie Galan]] of Trilogy Productions. The song can be heard on the [[High School Musical (album)|''High School Musical'' soundtrack]]. B5 also performed a new version of the classic [[Earth Wind & Fire]] song, "[[Shining Star]]" on Disney's [[Hannah Montana (album)|''Hannah Montana'' soundtrack]], also produced by [[Drew Lane]] and [[Eddie Galan]].

where

B5 appeared in the 2006 movie ''[[The Guardians]]'' and made a cameo appearance in [[Ciara]]'s "[[1, 2 Step]]" video and [[Ludacris]]' video, "[[Georgia (rap)|Georgia]]".

:<math>c = { 1 \over \sqrt{ \mu_o \varepsilon_o } } = 2.998 \times 10^8 </math> meters per second

As of January 2007, B5 are working on their second album, '''Don't Talk Just Listen''', which has been scheduled to be released on September 18, 2007. They were reported to be working with producers such as [[Pooh Bear]], [[Soul Diggaz]], [[Scott Storch]], [[Drew Lane]] and [[Eddie Galan]], [[Daron Jones]],[[Will.i.am]], Jim Jonsin, Mischka, [[Just Blaze]], Brian Cox, Underdogs, Insomniax, [[Mario Winans]], 88 Productions, Wiley Morris, Corte Ellis, Danja and many more. {{fact|date=March 2007}}

is the [[speed of light]] in [[free space]]. The magnetic [[permeability]] <math>\ \mu_o</math> and the electric [[permittivity]] <math>\ \varepsilon_o</math> are important [[physical constants]] that play a key role in [[electromagnetism | electromagnetic theory]].

==Lineup==

These equations derive from [[Maxwell's equations]] and describe the propagation of [[electromagnetic radiation|electromagnetic waves]] in a vacuum.

*Dustin Breeding - born [[October 8]],1987 in [[St. Petersburg, Florida]].Age 19

*Kelly Breeding - born [[February 17]],1989 in [[Liberal, Kansas]]. Age 18

*Patrick Breeding - born [[September 19]],1990 in [[Liberal, Kansas]]. Age 16

*Carnell Breeding - born [[November 30]],1991 in [[Bridgeport, Connecticut]]. Age 15

*Bryan Breeding - born [[October 14]], 1994 in [[Liberal, Kansas]]. Age 12

==Discography==

== Conservation of charge==

===Albums===

{| border="4" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-size: 95%;"

|- bgcolor="#CCCCCC" align="center"

!align="left"|Album Cover

!align="left"|Album Info

|-

|align="left"|[[Image:B5album1.jpg|150px]]

|align="left"|'''''[[B5 (album)|B5]]'''''

*Released: [[July 19]], [[2005]] (U.S.) #19 (Billboard 200)<br>#7(U.S.R&B/Hip Hop)

*Label: [[Bad Boy Records|Bad Boy]]

*U.S. Sales:

*RIAA certification: Gold

*Singles:

*U got Me

*All I do

|align="left"|[[Image:DontTalkJustListen.jpg|150px]]

|align="left"|'''''[[Don't Talk, Just Listen]]'''''

*Released: [[September 18]], [[2007]] (U.S.) TBA (Billboard 200) TBA (U.S.R&B/Hip Hop)

*Label: [[Bad Boy Records|Bad Boy]]

*U.S. Sales:

*RIAA certification:

*Singles:

**[[Hydrolics]] (Feat. [[Bow Wow]])

|}

[[Charge conservation | Conservation of charge]] requires that the time rate of change of the total charge density within an enclosed volume ''V'' must equal the net current density '''J''' flowing into the surface ''S'' enclosing the volume:

===Singles===

:<math> \int_S \mathbf{J} \cdot d \mathbf{a} = - {d \over dt} \int_V \rho \cdot dV</math>

*"All I Do"- peaked at #71 on [[Hot R&B/Hip-Hop Singles and Tracks]]

*"[[U Got Me]]"

*"Someday At Christmas"

*"Let's Groove"

*"Who's Afraid Of The Big Bad Wolf?" peaked at #82 on [[Pop 100]]

*"Getcha Head in the Game"

*"[[Hydrolics]]" featuring [[Bow Wow]] and [[Sean Combs|Diddy]]

==External links==

*[http://b5online.com B5's Official Site]

*[http://www.myspace.com/b5online B5's Official Myspace]

*[http://www.ohsob5.com B5's Official Fansite]

<br>

{{Bad boy}}

From the [[divergence theorem]], we can convert this relationship from integral form to differential form:

[[Category:2000s music groups]]

[[Category:Boy bands]]

[[Category:Family musical groups]]

[[Category:Georgia (U.S. state) musical groups]]

[[Category:Bad Boy Records artists]]

[[Category:Musical groups with siblings]]

[[Category:People from Atlanta]]

[[sv:B5 (band)]]

:<math> \nabla \cdot \mathbf{J} = - { d \rho \over dt } </math>

== Ampere's Law prior to Maxwell's correction ==

In its original form, [[Ampere's Law]] relates the magnetic field '''H''' to its source, the current density '''J''':

:<math> \oint_C \mathbf{H} \cdot d \mathbf{l} = \int_S \mathbf{J} \cdot d \mathbf{a}</math>

Again, we can convert to differential form, this time using [[Stoke's theorem]]:

:<math> \nabla \times \mathbf{H} = \mathbf{J} </math>

== Inconsistency between Ampere's Law and Conservation of Charge ==

Maxwell discovered an important inconsistency between Ampere's Law and the Conservation of Charge.

If we take the divergence of both sides of Ampere's Law, we find

:<math> \nabla \cdot ( \nabla \times \mathbf{H} ) = \nabla \cdot \mathbf{J} </math>

The divergence of the curl of any vector field &ndash; in this case, the magnetic field '''H''' &ndash; is always equal to zero:

:<math> \nabla \cdot ( \nabla \times \mathbf{H} ) = 0 </math>

Combining these two equations implies that

:<math>\nabla \cdot \mathbf{J} = 0 </math>

From the conservation of charge, we know that

:<math> \nabla \cdot \mathbf{J} = - { d \rho \over dt } </math>

Ampere's Law, then, implies that

:<math>{ d \rho \over dt } = 0</math>

This last result suggests that the net charge density at any point in space is a fixed constant that cannot ''ever'' change, which is of course absurd. Not only is this outcome contrary to all physical intuition, it also directly contradicts the empirical results of thousands of laboratory experiments. It requires not only that electrical charge is conserved, but that it cannot be re-distributed from one place to another. But we know that electrical currents can and do re-distribute electrical charge. As long as the ''total'' amount of charge remains constant, conservation of charge allows for the movement of charge from one place to another. So this last result is incorrect.

Something was clearly missing from Ampere's Law, and Maxwell figured out what it was.

==Maxwell's correction to Ampere's Law==

To understand Maxwell's correction to Ampere's Law, we need to look at another of Maxwell's Equations, namely, [[Gauss's Law]] in integral form:

:<math> \int_S \varepsilon_o \mathbf{E} \cdot d \mathbf{a} = \int_V \rho \cdot dV</math>

Again, using the divergence theorem, we can convert this equation to differential form:

:<math> \nabla \cdot \varepsilon_o \mathbf{E} = \rho </math>

Taking the derivative with respect to time of both sides, we find:

:<math> {\partial \over \partial t } ( \nabla \cdot \varepsilon_o \mathbf{E} ) = {\partial \rho \over \partial t}</math>

Reversing the order of differentiation on the left-hand side, we obtain

:<math> \nabla \cdot \varepsilon_o {\partial \mathbf{E} \over \partial t } = { \partial \rho \over \partial t}</math>

This last result, along with Ampere's Law and the conservation of charge equation, suggests that there are actually ''two'' sources of the magnetic field: the current density '''J''', as Ampere had already established, and the so-called '''displacement current''':

:<math> {\partial \mathbf{D} \over \partial t } = \varepsilon_o {\partial \mathbf{E} \over \partial t } </math>

So the corrected form of Ampere's Law, which Maxwell discovered, becomes:

:<math> \nabla \times \mathbf{H} = \mathbf{J} + \varepsilon_o {\partial \mathbf{E} \over \partial t } </math>

== Maxwell's discovery that light is an electromagnetic wave==

Maxwell's correction of Ampere's Law set the stage for an even more important and, at the time, startling discovery. Maxwell realized that the equations of electromagnetism suggest that electric and magnetic fields can propagate through free space &ndash; in other words, in the '''absence''' of matter &ndash; as electromagnetic waves, and further, that the speed of propagation of these waves is exactly the same as the [[speed of light]]. Upon making this discovery in [[1865]], Maxwel wrote:

:''This velocity is so nearly [[speed of light|that of light]], that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.''

To obtain [[Electromagnetic waves | electromagnetic waves]] in a vacuum note that Maxwell's equations in a vacuum are

:<math> \nabla \cdot \mathbf{E} = 0</math>

:<math> \nabla \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}} {\partial t}</math>

:<math> \nabla \cdot \mathbf{H} = 0</math>

:<math> \nabla \times \mathbf{H} =\varepsilon_0 \frac{ \partial \mathbf{E}} {\partial t}</math>

If we take the curl of these equations and note the vector identity

:<math>\nabla \times \left( \nabla \times \mathbf{A} \right) = \nabla \left( \nabla \cdot \mathbf{A} \right) - \nabla^2 \mathbf{A}</math>

we recover the wave equations, which describe wave propagation at speed c.

==See also==

* [[Maxwell's equations]]

* [[Electromagnetic waves]]

* [[Charge conservation]]

* [[Light]]

* [[Albert Einstein]]

* [[James Clerk Maxwell]]

* [[Michael Faraday]]

* [[Heinrich Hertz]]

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